**4.2. The luminosity distance and gravitational lensing**

Before proceeding to discuss possible constraints on
_{} from gravitational
lensing in this section and high redshift supernovae in the next,
let us introduce a quantity which plays a crucial role in these discussions,
namely the *luminosity distance* *d*_{L}(*z*)
up to a given redshift *z*. Consider an object of absolute luminosity
located at a coordinate
distance *r* from an observer at *r* = 0. Light emitted
by the object at a time *t* is received by the observer at *t*
= *t*_{0},
*t* and *t*_{0} being related by the cosmological
redshift 1 + *z* = *a*(*t*_{0}) /
*a*(*t*). The luminosity flux reaching the observer is

(22) |

where *d*_{L} is the luminosity distance to the object
[142]

(23) |

The luminosity distance *d*_{L} depends sensitively
upon both the spatial curvature and the
expansion dynamics of the universe. To demonstrate this we
determine *d*_{L} using the expression for the coordinate
distance *r*
obtained by setting *ds*^{2} = 0 in (2), resulting in

(24) |

which gives

(25) |

where =
_{0}^{t}
*cdt* / *a*(*t*).

Furthermore, since
*dz* / *dt* = - (1 + *z*)*H*(*z*), we get

(26) |

where *h*(*z*) = *H*(*z*) / *H*_{0} is
defined in (18), and, in a universe with several components

(27) |

Substituting (27) and (26) in (23) we get the following expression for the luminosity distance in a multicomponent universe with a cosmological term [26]

(28) |

where

(29) |

and *S*(*x*) is defined as follows:
*S*(*x*) = sin(*x*) if
= 1
(_{total}
> 1), *S*(*x*) = sinh(*x*) if
= - 1
(_{total}
< 1), *S*(*x*) = *x* if
= 0
(_{total} = 1).

Before we turn to applications, let us consider a simple
example which
provides us with an insight into the role played by the
luminosity distance *d*_{L} in cosmology.
In a spatially flat universe the expression for *d*_{L}
simplifies considerably,
so that we get for the matter dominated model (*a*
*t*^{2/3}):

(30) |

On the other hand in de Sitter space (*a*
exp(*H*_{0} *t*))

(31) |

Comparing (30) and (31) we find
*d*_{L}^{DS}(*z*) >
*d*_{L}^{MD}(*z*),
which means that an object located
at a fixed redshift will appear brighter in
an Einstein-de Sitter universe than it will in de Sitter space
(equivalently in the steady state model).
^{(5)}
This is also true for a
two component universe consisting of matter and a cosmological constant
as demonstrated in Fig 4. In a
spatially flat universe the presence of a
-term increases the
luminosity distance to a given redshift, leading to interesting
astrophysical
consequences. Since the physical volume associated with a unit redshift
interval increases in models with
> 0, the
likelihood that light from a quasar
will encounter a lensing galaxy is larger in such models. Consequently
the probability that a quasar is lensed by
intervening galaxies increases appreciably in a
dominated
universe, and can be used as a test to constrain the value of
_{}
[72,
71,
192].
Following
[73,
26,
37]
we give below the probability of a quasar
at redshift *z*_{s} being lensed relative to the fiducial
Einstein-de Sitter model
(_{m} = 1)

(32) |

where *d* (*z*_{1}, *z*_{2}) is a
generalization of the angular distance
*d*_{A} = *d*_{L}(1 + *z*)^{-2}
discussed in Section 4.5:

(33) |

where

(34) |

and *S*(_{12}) is defined as follows,
*S*(_{12}) =
sin(_{12}) if
= 1
(_{total}
> 1), *S*(_{12}) =
sinh(_{12}) if
= -1
(_{total}
< 1), *S*(_{12}) =
_{12}
if = 0
(_{total} = 1).
In Fig 5 we show the lensing probability
*P*(*lens*) for the spatially flat universe
_{m} +
_{} = 1.
A large increase in the lensing probability over the fiducial
_{m} = 1
value is clearly seen in models with low
_{m}
(high _{}).
(For a broader analysis of parameter space see
[26].)

Turning now to the observational situation,
at the time of writing the best observational estimates give a
2 upper bound
_{} < 0.66
obtained from multiple images of lensed quasars
[111,
112,
136].
Since radio sources are not plagued by some of the systematic
errors arising in an optical search (notably extinction in the lens
galaxy and the quasar discovery process) a search involving radio
selected lenses can yield useful complementary information to optical
searches [60].
Recent work by Falco et al (1998) gives
_{} < 0.73 which
is only marginally consistent with optical estimates,
a combined analysis of optical and radio data yields a slightly more
conservative upper bound
_{} < 0.62 at the
2 level (for flat
universes)
[60].
(Constraints on
_{} from both
lensing and Type 1a Supernovae are discussed in
[201];
also see next section. An interesting new method of constraining
_{} from weak
lensing in clusters is discussed in
[67],
also see
[12]
and section 4.6.)
Improved understanding of statistical and systematic uncertainties combined
with new surveys and better quality data promise to make gravitational
lensing a powerful technique for constraining cosmological parameters and
cosmological world models.

^{5} For instance a galaxy at
redshift *z* = 3 will appear 9 times brighter in a flat matter
dominated universe than it will in de Sitter space (see
Fig 4).
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